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Theorem elpcliN 35497
Description: Implication of membership in the projective subspace closure function. (Contributed by NM, 13-Sep-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
elpcli.s 𝑆 = (PSubSp‘𝐾)
elpcli.c 𝑈 = (PCl‘𝐾)
Assertion
Ref Expression
elpcliN (((𝐾𝑉𝑋𝑌𝑌𝑆) ∧ 𝑄 ∈ (𝑈𝑋)) → 𝑄𝑌)

Proof of Theorem elpcliN
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 simp1 1081 . . . . . 6 ((𝐾𝑉𝑋𝑌𝑌𝑆) → 𝐾𝑉)
2 simp2 1082 . . . . . . 7 ((𝐾𝑉𝑋𝑌𝑌𝑆) → 𝑋𝑌)
3 eqid 2651 . . . . . . . . 9 (Atoms‘𝐾) = (Atoms‘𝐾)
4 elpcli.s . . . . . . . . 9 𝑆 = (PSubSp‘𝐾)
53, 4psubssat 35358 . . . . . . . 8 ((𝐾𝑉𝑌𝑆) → 𝑌 ⊆ (Atoms‘𝐾))
653adant2 1100 . . . . . . 7 ((𝐾𝑉𝑋𝑌𝑌𝑆) → 𝑌 ⊆ (Atoms‘𝐾))
72, 6sstrd 3646 . . . . . 6 ((𝐾𝑉𝑋𝑌𝑌𝑆) → 𝑋 ⊆ (Atoms‘𝐾))
8 elpcli.c . . . . . . 7 𝑈 = (PCl‘𝐾)
93, 4, 8pclvalN 35494 . . . . . 6 ((𝐾𝑉𝑋 ⊆ (Atoms‘𝐾)) → (𝑈𝑋) = {𝑧𝑆𝑋𝑧})
101, 7, 9syl2anc 694 . . . . 5 ((𝐾𝑉𝑋𝑌𝑌𝑆) → (𝑈𝑋) = {𝑧𝑆𝑋𝑧})
1110eleq2d 2716 . . . 4 ((𝐾𝑉𝑋𝑌𝑌𝑆) → (𝑄 ∈ (𝑈𝑋) ↔ 𝑄 {𝑧𝑆𝑋𝑧}))
12 elintrabg 4521 . . . . 5 (𝑄 {𝑧𝑆𝑋𝑧} → (𝑄 {𝑧𝑆𝑋𝑧} ↔ ∀𝑧𝑆 (𝑋𝑧𝑄𝑧)))
1312ibi 256 . . . 4 (𝑄 {𝑧𝑆𝑋𝑧} → ∀𝑧𝑆 (𝑋𝑧𝑄𝑧))
1411, 13syl6bi 243 . . 3 ((𝐾𝑉𝑋𝑌𝑌𝑆) → (𝑄 ∈ (𝑈𝑋) → ∀𝑧𝑆 (𝑋𝑧𝑄𝑧)))
15 sseq2 3660 . . . . . . . 8 (𝑧 = 𝑌 → (𝑋𝑧𝑋𝑌))
16 eleq2 2719 . . . . . . . 8 (𝑧 = 𝑌 → (𝑄𝑧𝑄𝑌))
1715, 16imbi12d 333 . . . . . . 7 (𝑧 = 𝑌 → ((𝑋𝑧𝑄𝑧) ↔ (𝑋𝑌𝑄𝑌)))
1817rspccv 3337 . . . . . 6 (∀𝑧𝑆 (𝑋𝑧𝑄𝑧) → (𝑌𝑆 → (𝑋𝑌𝑄𝑌)))
1918com13 88 . . . . 5 (𝑋𝑌 → (𝑌𝑆 → (∀𝑧𝑆 (𝑋𝑧𝑄𝑧) → 𝑄𝑌)))
2019imp 444 . . . 4 ((𝑋𝑌𝑌𝑆) → (∀𝑧𝑆 (𝑋𝑧𝑄𝑧) → 𝑄𝑌))
21203adant1 1099 . . 3 ((𝐾𝑉𝑋𝑌𝑌𝑆) → (∀𝑧𝑆 (𝑋𝑧𝑄𝑧) → 𝑄𝑌))
2214, 21syld 47 . 2 ((𝐾𝑉𝑋𝑌𝑌𝑆) → (𝑄 ∈ (𝑈𝑋) → 𝑄𝑌))
2322imp 444 1 (((𝐾𝑉𝑋𝑌𝑌𝑆) ∧ 𝑄 ∈ (𝑈𝑋)) → 𝑄𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  {crab 2945  wss 3607   cint 4507  cfv 5926  Atomscatm 34868  PSubSpcpsubsp 35100  PClcpclN 35491
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-psubsp 35107  df-pclN 35492
This theorem is referenced by:  pclfinclN  35554
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